Cops and robbers for hyperbolic and virtually free groups

TL;DR

The paper introduces two group invariants, the strong cop number $sCop$ and the weak cop number $wCop$, defined via cops-and-robbers games on Cayley graphs and shown to be invariant under quasi-isometries. It proves sharp geometric characterizations: $G$ is Gromov-hyperbolic iff $sCop(G)=1$, and $G$ is virtually free iff $wCop(G)=1$, with $sCop(G)\le wCop(G)$ and baseline cases like $sCop(\mathbb{Z})=wCop(\mathbb{Z})=1$. A meta-gaming framework using sequences of quasi-homotheties is developed to upgrade weak strategies to strong ones, enabling explicit determinations for families such as lamplighter groups, Baumslag-Solitar groups, and Thompson's group $F$. The framework yields that several natural groups have infinite strong cop numbers, and it rules out intermediate cop numbers in CAT(0)-groups, while also providing tools and open questions about cop-number behavior in broader geometric contexts."

Abstract

Lee, Martínez-Pedroza and Rodríguez-Quinche define two new group invariants, the strong cop number $\operatorname{sCop}$ and the weak cop number $\operatorname{wCop}$, by examining winning strategies for certain combinatorial games played on Cayley graphs of finitely generated groups. We show that a finitely generated group $G$ is Gromov-hyperbolic if and only if $\operatorname{sCop(G)} = 1$. We show that $G$ is virtually free if and only if $\operatorname{wCop(G)}=1$, answering a question by Cornect and Martínez-Pedroza. We show that $\operatorname{sCop}(\mathbb{Z}^2) = \infty$, answering a question from the original paper. It is still unknown whether there exist finite cop numbers not equal to 1, but we show that this is not possible for CAT(0)-groups. We provide machinery to explicitly compute strong cop numbers and give examples by applying it to certain lamplighter groups, the solvable Baumslag-Solitar groups, and Thompson's group F.

Figures (10)

• Figure 1: An overview of known results for the strong and weak cop numbers. The existence of groups with combinations of cop numbers marked with ? is an open question. The marking $?^\star$ indicates that no group is known, but graphs are known.
• Figure 2: The robber finds a $\delta$-thin, but not $(\delta-1)$-thin bigon and waits at $p$. Once the cop gets closer than $5\lambda$, the robber moves from $p$ to $p'$ along one of two possible paths $\eta_+$ or $\eta_-$, at most one of which can be blocked by the cop $c_0$.
• Figure 3: To prove that the robber $r$ is not caught by the cop $c$ during its movement along $\eta_+$ we consider five cases. Case (e) is only needed if $\ell < t+\delta$.
• Figure 4: The negation of the Bottleneck property guarantees the existence of a path $\gamma$ connecting $x$ and $y$ outside a ball of size $6\lambda$ around $y$.
• Figure 5: At the beginning of a meta-stage, the cop's position $c_0$ is provided to the oracle $\Delta$ via $\pi$. In the oracle there is a winning strategy by following along the path $\overline{p}_1, \ldots , \overline{p}_n$, which the robber implements in $\Gamma$, by following along $p_1 := \iota(\overline{p}_1), \ldots , p_n$.

Theorems & Definitions (48)

• Theorem 1.1: \ref{['thm:scop-hyperbolic']}
• Theorem 1.2: \ref{['thm:wcop-virtfree']}
• Theorem 1.3: \ref{['thm:scop-Zn']}
• Theorem 1.4: \ref{['thm:scop-lamplighter']}
• Theorem 1.5: \ref{['thm:scop-bs']}
• Theorem 1.6: Meta-gaming theorem, for details see \ref{['thm:meta-gaming']} and \ref{['cor:self-meta-gaming']}
• Corollary 1.7: \ref{['cor:no-homothety-in-hyp-groups']}
• Corollary 1.8: \ref{['cor:no-homothety-in-hyp-plane']}
• Proposition 1.9: Proposition \ref{['prop:intermediate-CAT0']}
• Remark 2.1